SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.9 → 1.6

Time: 15.5s

Precision: binary64

Cost: 19904

Math

\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

↓

\[\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))

↓

(FPCore (x y z t)
 :precision binary64
 (fma z (* y (- (tanh (/ t y)) (tanh (/ x y)))) x))

C

double code(double x, double y, double z, double t) {
	return x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
}

↓

double code(double x, double y, double z, double t) {
	return fma(z, (y * (tanh((t / y)) - tanh((x / y)))), x);
}

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * z) * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))))
end

↓

function code(x, y, z, t)
	return fma(z, Float64(y * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y * z), $MachinePrecision] * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(z * N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Target

Original	4.9
Target	2.1
Herbie	1.6

\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

Derivation

1. Initial program 4.9

   \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

2. Simplified 1.6

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]
   Proof

   [Start] 4.9	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   +-commutative [=>] 4.9	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
   *-commutative [=>] 4.9	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
   associate-*l* [=>] 1.6	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
   fma-def [=>] 1.6	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

3. Final simplification 1.6

   \[\leadsto \mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right) \]

Alternatives

Alternative 1
Error	2.1
Cost	13632

\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

Alternative 2
Error	22.8
Cost	7638

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+252}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.48 \cdot 10^{+91} \lor \neg \left(z \leq 2.3 \cdot 10^{+26}\right) \land \left(z \leq 1.15 \cdot 10^{+88} \lor \neg \left(z \leq 1.6 \cdot 10^{+194}\right)\right):\\ \;\;\;\;z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	9.9
Cost	7497

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-65} \lor \neg \left(t \leq 3.7 \cdot 10^{-134}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	21.6
Cost	7376

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-258}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-98}:\\ \;\;\;\;x + \frac{z}{\frac{-y}{y \cdot x}}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+48}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	11.4
Cost	7241

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-144} \lor \neg \left(t \leq 1.3 \cdot 10^{-128}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tanh \left(\frac{x}{y}\right) \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 6
Error	13.9
Cost	7240

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+252}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;z \cdot \left(y \cdot t_1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_1 \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 7
Error	15.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+105}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \end{array} \]

Alternative 8
Error	15.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+105} \lor \neg \left(y \leq 1.15 \cdot 10^{-43}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	23.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -2.16 \cdot 10^{-305}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-96}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	23.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023040 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :herbie-target
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))